Intersection of Two Lines

We are given two lines \({L_1}\) and \({L_2}\) , and we are required to find the point of intersection (if they are non-parallel) and the angle at which they are inclined to one another, i.e., the angle of intersection. Evaluating the point of intersection is a simple matter of solving two simultaneous linear equations. Let the equations of the two lines be (written in the general form):

Conventionally, we would be interested only in the acute angle between the two lines and thus we have to have \(\tan \theta \) as a positive quantity. So in the expression above, if the expression \(\frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}\) turns out to be negative, this would be the tangent of the obtuse angle between the two lines; thus, to get the acute angle between the two lines, we use the magnitude of this expression.

From this relation, we can easily deduce the conditions on \({m_1}\) and \({m_2}\) such that the two lines \({L_1}\) and \({L_2}\) are parallel or perpendicular.

If the lines are parallel, \(\theta = 0\) , so that \({m_1} = {m_2}\) , which is intuitively obvious since parallel lines must have the same slope.

For the two lines to be perpendicular, \(\theta = \frac{\pi }{2}\), so that \(\cot \theta = 0\); this can happen if \(1 + {m_1}{m_2} = 0\) or \({m_1}{m_2} = - 1\) .

If the lines \({L_1}\) and \({L_2}\) are given in the general form given in the general form \(ax + by + c = 0\), the slope of this line is \(m = - \frac{a}{b}\). Thus, the condition for \({L_1}\) and \({L_2}\) to be parallel is: